NCERT CLASS 6 MATHS KNOWING OUR NUMBERS TEXTBOOK

1. Counting things is easy for us now. We can count objects in large numbers,
for example, the number of students in the school, and represent them
through numerals. We can also communicate large numbers using suitable
number names.
It is not as if we always knew how to convey large quantities in conversation
or through symbols. Many thousands years ago, people knew only small
numbers. Gradually, they learnt how to handle larger numbers.They also learnt
how to express large numbers in symbols. All this came through collective
efforts of human beings. Their path was not easy, they struggled all along the
way. In fact, the development of whole of Mathematics can be understood
this way.As human beings progressed, there was greater need for development
of Mathematics and as a result Mathematics grew further and faster.
We use numbers and know many things about them. Numbers help us
count concrete objects. They help us to say which collection of objects
is bigger and arrange them in order e.g., first, second, etc. Numbers are
used in many different contexts and in many ways. Think about various
situations where we use numbers. List five distinct situations in which
numbers are used.
We enjoyed working with numbers in our previous classes. We have added,
subtracted, multiplied and divided them.We also looked for patterns in number
sequences and done many other interesting things with numbers. In this chapter,
we shall move forward on such interesting things with a bit of review and
revision as well.
1.1 Introduction
Chapter1Chapter1Chapter1Chapter1Chapter1
Knowing ourKnowing ourKnowing ourKnowing ourKnowing our
NumbersNumbersNumbersNumbersNumbers

2. MATHEMATICS
2
1.2 Comparing Numbers
As we have done quite a lot of this earlier, let us see if we remember which is
the greatest among these :
(i) 92, 392, 4456, 89742
(ii) 1902, 1920, 9201, 9021, 9210
So, we know the answers.
Discuss with your friends, how you find the number that is the greatest.
Can you instantly find the greatest and the smallest numbers in each row?
1. 382, 4972, 18, 59785, 750. Ans. 59785 is the greatest and
18 is the smallest.
2. 1473, 89423, 100, 5000, 310. Ans. ____________________
3. 1834, 75284, 111, 2333, 450 . Ans. ____________________
4. 2853, 7691, 9999, 12002, 124. Ans. ____________________
Was that easy? Why was it easy?
We just looked at the number of digits and found the answer.
The greatest number has the most thousands and the smallest is
only in hundreds or in tens.
Make five more problems of this kind and give to your friends
to solve.
Now, how do we compare 4875 and 3542?
This is also not very difficult.These two numbers have the
same number of digits. They are both in thousands. But the digit
at the thousands place in 4875 is greater than that in 3542.
Therefore, 4875 is greater than 3542.
Next tell which is greater, 4875 or
4542? Here too the numbers have the
same number of digits. Further, the digits
at the thousands place are same in both.
What do we do then? We move to the
next digit, that is to the digit at the
hundreds place.The digit at the hundreds
place is greater in 4875 than in 4542.
Therefore, 4875 is greater than 4542.
Find the greatest and the smallest
numbers.
(a) 4536, 4892, 4370, 4452.
(b) 15623, 15073, 15189, 15800.
(c) 25286, 25245, 25270, 25210.
(d) 6895, 23787, 24569, 24659.

3. KNOWING OUR NUMBERS
3
9 8 6 7
1 0 2 7
4
4
9
9
1
If the digits at hundreds place are also same in the two numbers, then what
do we do?
Compare 4875 and 4889 ; Also compare 4875 and 4879.
1.2.1 How many numbers can you make?
Suppose, we have four digits 7, 8, 3, 5. Using these digits we want to make
different 4-digit numbers such that no digits are repeated in a number. Thus,
7835 is allowed, but 7735 is not. Make as many 4-digit numbers as you can.
Which is the greatest number you can get? Which is the smallest number?
The greatest number is 8753 and the smallest is 3578.
Think about the arrangement of the digits in both. Can you say how the largest
number is formed? Write down your procedure.
1. Use the given digits without repetition and make the greatest and smallest
4-digit numbers.
(a) 2, 8, 7, 4 (b) 9, 7, 4, 1 (c) 4, 7, 5, 0
(d) 1, 7, 6, 2 (e) 5, 4, 0, 3
(Hint : 0754 is a 3-digit number.)
2. Now make the greatest and the smallest 4-digit numbers by using any one
digit twice.
(a) 3, 8, 7 (b) 9, 0, 5 (c) 0, 4, 9 (d) 8, 5, 1
(Hint : Think in each case which digit will you use twice.)
3. Make the greatest and the smallest 4-digit numbers using any four different
digits with conditions as given.
(a) Digit 7 is always at Greatest
ones place
Smallest
(Note, the number cannot begin with the digit 0. Why?)
(b) Digit 4 is always Greatest
at tens place
Smallest
(c) Digit 9 is always at Greatest
hundreds place
Smallest
(d) Digit 1 is always at Greatest
thousands place
Smallest 1

4. MATHEMATICS
4
Ramhari
(160 cm)
Dolly
(154 cm)
Mohan
(158 cm)
Shashi
(159 cm)
Rs 2635 Rs 1897 Rs 2854 Rs 1788 Rs 3975
4. Take two digits, say 2 and 3. Make 4-digit numbers using both the digits equal
numberoftimes.
Whichisthegreatestnumber?
Whichisthesmallestnumber?
Howmanydifferentnumberscanyoumakeinall?
Stand in proper order
1. Who is the tallest?
2. Who is the shortest?
(a) Can you arrange them in the increasing order of their heights?
(b) Can you arrange them in the decreasing order of their heights?
Which to buy?
SohanandRitawent
to buy an almirah.
There were many
almirahs available
withtheirpricetags.
(a) Can you arrange their prices in increasing
order?
(b) Can you arrange their prices in
decreasing order?
Ascending order Ascending order means arrangement from the smallest to the
greatest.
Descending order Descending order means arrangement from the greatest to
the smallest.
Think of five more situations
where you compare three or
more quantities.

5. KNOWING OUR NUMBERS
5
1. Arrange the following numbers in ascending order :
(a) 847, 9754, 8320, 571 (b) 9801, 25751, 36501, 38802
2. Arrange the following numbers in descending order :
(a) 5000, 7500, 85400, 7861 (b) 1971, 45321, 88715, 92547
Make ten such examples of ascending/descending order and solve them.
1.2.2 Shifting digits
Have you thought what fun it would be if the digits in a number could shift
(move) from one place to the other?
Think about what would happen to 182. It could become as large as 821
and as small as 128. Try this with 391 as well.
Now think about this. Take any 3-digit number and exchange the digit at
the hundreds place with the digit at the ones place.
(a) Is the new number greater than the former one?
(b) Is the new number smaller than the former number?
Write the numbers formed in both ascending and descending order.
Before 7 9 5
Exchanging the 1st and the 3rd tiles.
After 5 9 7
If you exchange the 1st and the 3rd tiles (i.e. digits), in which case does the
number become greater? In which case does it become smaller?
Try this with a 4-digit number.
1.2.3 Introducing 10,000
We know that beyond 99 there is no 2-digit number. 99 is the greatest 2-digit
number. Similarly, the greatest 3-digit number is 999 and the greatest 4-digit
number is 9999. What shall we get if we add 1 to 9999?
Look at the pattern : 9 + 1 = 10 = 10 × 1
99 + 1 = 100 = 10 × 10
999 + 1 = 1000 = 10 × 100
We observe that
Greatest single digit number + 1 = smallest 2-digit number
Greatest 2-digit number + 1 = smallest 3-digit number
Greatest 3-digit number + 1 = smallest 4-digit number

6. MATHEMATICS
6
Should not we then expect that on adding 1 to the greatest 4-digit number,
we would get the smallest 5-digit number, that is 9999 + 1 = 10000.
The new number which comes next to 9999 is 10000. It is called
ten thousand. Further, we expect 10000 = 10 × 1000.
1.2.4 Revisiting place value
You have done this quite earlier, and you will certainly remember the expansion
of a 2-digit number like 78 as
78 = 70 + 8 = 7 × 10 + 8
Similarly, you will remember the expansion of a 3-digit number like 278 as
278 = 200 + 70 + 8 = 2 × 100 + 7 × 10 + 8
We say, here, 8 is at ones place, 7 is at tens place and 2 at hundreds place.
Later on we extended this idea to 4-digit numbers.
For example, the expansion of 5278 is
5278 = 5000 + 200 + 70 + 8
= 5 × 1000 + 2 × 100 + 7 × 10 + 8
Here, 8 is at ones place, 7 is at tens place, 2 is at hundreds place and 5 is at
thousands place.
With the number 10000 known to us, we may extend the idea further. We
may write 5-digit numbers like
45278 = 4 × 10000 + 5 × 1000 + 2 × 100 + 7 × 10 + 8
We say that here 8 is at ones place, 7 at tens place, 2 at hundreds place,
5 at thousands place and 4 at ten thousands place. The number is read as forty
five thousand, two hundred seventy eight. Can you now write the smallest
and the greatest 5-digit numbers?
Read and expand the numbers wherever there are blanks.
Number Number Name Expansion
20000 twenty thousand 2 × 10000
26000 twenty six thousand 2 × 10000 + 6 × 1000
38400 thirty eight thousand 3 × 10000 + 8 × 1000
four hundred + 4 × 100
65740 sixty five thousand 6 × 10000 + 5 × 1000
seven hundred forty + 7 × 100 + 4 × 10

7. KNOWING OUR NUMBERS
7
1.2.5 Introducing 1,00,000
Which is the greatest 5-digit number?
Adding 1 to the greatest 5-digit number, should give the smallest
6-digit number : 99,999 + 1 = 1,00,000
This number is named one lakh. One lakh comes next to 99,999.
10 × 10,000 = 1,00,000
We may now write 6-digit numbers in the expanded form as
2,46,853 = 2 × 1,00,000 + 4 × 10,000 + 6 × 1,000 +
8 × 100 + 5 × 10 +3 × 1
This number has 3 at ones place, 5 at tens place, 8 at hundreds place, 6 at
thousands place, 4 at ten thousands place and 2 at lakh place. Its number
name is two lakh forty six thousand eight hundred fifty three.
Read and expand the numbers wherever there are blanks.
Number Number Name Expansion
3,00,000 three lakh 3 × 1,00,000
3,50,000 three lakh fifty thousand 3 × 1,00,000 + 5 × 10,000
3,53,500 three lakh fifty three 3 × 1,00,000 + 5 × 10,000
thousand five hundred + 3 × 1000 + 5 × 100
4,57,928 _______________ _______________
4,07,928 _______________ _______________
4,00,829 _______________ _______________
4,00,029 _______________ _______________
89324 eighty nine thousand 8 × 10000 + 9 × 1000
three hundred twenty four + 3 × 100 + 2 × 10 + 4 × 1
50000 _______________ _______________
41000 _______________ _______________
47300 _______________ _______________
57630 _______________ _______________
29485 _______________ _______________
29085 _______________ _______________
20085 _______________ _______________
20005 _______________ _______________
Write five more 5-digit numbers, read them and expand them.

8. MATHEMATICS
8
1. Give five examples where the number of things counted would be more than
6-digit number.
2. Starting from the greatest 6-digit number, write the previous five numbers in
descending order.
3. Starting from the smallest 8-digit number, write the next five numbers in
ascending order and read them.
1.2.6 Larger numbers
If we add one more to the greatest 6-digit number we get the smallest 7-digit
number. It is called ten lakh.
Write down the greatest 6-digit number and the smallest 7-digit number.
Write the greatest 7-digit number and the smallest 8-digit number. The smallest
8-digit number is called one crore.
Complete the pattern :
9 + 1 = 10
99 + 1 = 100
999 + 1 = _______
9,999 + 1 = _______
99,999 + 1 = _______
9,99,999 + 1 = _______
99,99,999 + 1 = 1,00,00,000
We come across large numbers in
many different situations.
For example, while the number of
children in your class would be a
2-digit number, the number of
children in your school would be
a 3 or 4-digit number.
The number of people in the nearby town would be much larger.
Is it a 5 or 6 or 7-digit number?
Do you know the number of people in your state?
How many digits would that number have?
Whatwouldbethenumberofgrainsinasackfullofwheat?A5-digitnumber,
a 6-digit number or more?
Remember
1 hundred = 10 tens
1 thousand = 10 hundreds
= 100 tens
1 lakh = 100 thousands
= 1000 hundreds
1 crore = 100 lakhs
= 10,000 thousands
1. What is 10 – 1 =?
2. What is 100 – 1 =?
3. What is 10,000 – 1 =?
4. What is 1,00,000 – 1 =?
5. What is 1,00,00,000 – 1 =?
(Hint : Use the said pattern.)

9. KNOWING OUR NUMBERS
9
1.2.7 An aid in reading and writing large numbers
Try reading the following numbers :
(a) 279453 (b) 5035472
(c) 152700375 (d) 40350894
Was it difficult?
Did you find it difficult to keep track?
Sometimes it helps to use indicators to read and write large numbers.
Shagufta uses indicators which help her to read and write large numbers.
Her indicators are also useful in writing the expansion of numbers. For example,
she identifies the digits in ones place, tens place and hundreds place in 257 by
writing them under the tables O, T and H as
H T O Expansion
2 5 7 2 × 100 + 5 × 10 + 7 × 1
Similarly, for 2902,
Th H T O Expansion
2 9 0 2 2 × 1000 + 9 × 100 + 0 × 10 + 2 × 1
One can extend this idea to numbers upto lakh as seen in the following
table. (Let us call them placement boxes). Fill the entries in the blanks left.
Number TLakh Lakh TTh Th H T O Number Name Expansion
7,34,543 — 7 3 4 5 4 3 Seven lakh thirty
-----------------
four thousand five
hundred forty three
32,75,829 3 2 7 5 8 2 9
---------------------
3 × 10,00,000
+ 2 × 1,00,000
+ 7 × 10,000
+ 5 × 1000
+ 8 × 100
+ 2 × 10 + 9
Similarly, we may include numbers upto crore as shown below :
Number TCr Cr TLakh Lakh TTh Th H T O Number Name
2,57,34,543 — 2 5 7 3 4 5 4 3 ...................................
65,32,75,829 6 5 3 2 7 5 8 2 9 Sixty five crore thirty
two lakh seventy five
thousand eight hundred
twenty nine
You can make other formats of tables for writing the numbers in expanded form.

10. MATHEMATICS
10
Use of commas
You must have noticed that in writing large numbers in the
sectionsabove,wehaveoftenusedcommas.Commashelpus
in reading and writing large numbers. In our Indian System
of Numeration we use ones, tens, hundreds, thousands and
then lakhs and crores. Commas are used to mark thousands,
lakhsandcrores.Thefirstcommacomesafterhundredsplace(threedigitsfromthe
right)andmarksthousands.Thesecondcommacomestwodigitslater(fivedigits
fromtheright).Itcomesaftertenthousandsplaceandmarkslakh.Thethirdcomma
comes after anothertwodigits(seven digitsfrom the right).Itcomesaftertenlakh
placeandmarkscrore.
Forexample, 5, 08, 01, 592
3, 32, 40, 781
7, 27, 05, 062
Tryreadingthenumbersgivenabove.Writefivemorenumbersinthisformand
readthem.
International System of Numeration
IntheInternationalSystemofNumeration,asitisbeingusedwehaveones,
tens, hundreds, thousands and then millions. One million is a thousand
thousands.Commasareusedtomarkthousandsandmillions.Itcomesafter
every three digits from the right. The first comma marks thousands and the
nextcommamarksmillions.Forexample,thenumber50,801,592isreadin
the International System as fifty million eight hundred one thousand five
hundredninetytwo.IntheIndianSystem,itisfivecroreeightlakhonethousand
five hundred ninety two.
Howmanylakhsmakeamillion?
Howmanymillionsmakeacrore?
Takethreelargenumbers.ExpresstheminbothIndianandInternational
Numerationsystems.
Thismayinterestyou:
Toexpressnumberslargerthanamillion,abillionisusedintheInternational
SystemofNumeration:1billion=1000million.
While writing
number names,
we do not use
commas.

11. KNOWING OUR NUMBERS
11
How much was the increase during
1991-2001? Try to find out.
Do you know what is India’s population
today? Try to find this too.
1. Read these numbers. Write them using placement boxes and then write
their expanded forms.
(i) 475320 (ii) 9847215 (iii) 97645310 (iv) 30458094
(a) Which is the smallest number?
(b) Which is the greatest number?
(c) Arrange these numbers in ascending and descending orders.
2. Read these numbers.
(i) 527864 (ii) 95432 (iii) 18950049 (iv) 70002509
(a) Write these numbers using placement boxes and then using commas in
Indian as well as International System of Numeration..
(b) Arrange these in ascending and descending order.
3. Take three more groups of large numbers and do the exercise given above.
Do you know?
India’s population increased by
27 million during 1921-1931;
37 million during 1931-1941;
44 million during 1941-1951;
78 million during 1951-1961!
Can you help me write the numeral?
To write the numeral for a number you can follow the boxes again.
(a) Forty two lakh seventy thousand eight.
(b) Two crore ninety lakh fifty five thousand eight hundred.
(c) Seven crore sixty thousand fifty five.
1. You have the following digits 4, 5, 6, 0, 7 and 8. Using them, make five
numbers each with 6 digits.
(a) Put commas for easy reading.
(b) Arrange them in ascending and descending order.
2. Take the digits 4, 5, 6, 7, 8 and 9. Make any three numbers each with 8
digits. Put commas for easy reading.
3. From the digits 3, 0 and 4, make five numbers each with 6 digits. Use commas.

12. MATHEMATICS
12
1. How many
centimetres make a
kilometre?
2. Name five large cities
in India. Find their
population.Also, find
the distance in
kilometres between
each pair of these cities.
1.3 Large Numbers in Practice
In earlier classes, we have learnt that we use centimetre (cm) as a unit of length.
For measuring the length of a pencil, the width of a book or
notebooks etc., we use centimetres. Our ruler has marks on each centimetre.
For measuring the thickness of a pencil, however, we find centimetre too big.
We use millimetre (mm) to show the thickness of a pencil.
(a) 10 millimetres = 1 centimetre
To measure the length of the classroom or
the school building, we shall find
centimetre too small. We use metre for the
purpose.
(b) 1 metre = 100 centimetres
= 1000 millimetres
Even metre is too small, when we have to
state distances between cities, say, Delhi
and Mumbai, or Chennai and Kolkata. For
this we need kilometres (km).
EXERCISE 1.1
1. Fill in the blanks:
(a) 1 lakh = _______ ten thousand.
(b) 1 million = _______ hundred thousand.
(c) 1 crore = _______ ten lakh.
(d) 1 crore = _______ million.
(e) 1 million = _______ lakh.
2. Place commas correctly and write the numerals:
(a) Seventy three lakh seventy five thousand three hundred seven.
(b) Nine crore five lakh forty one.
(c) Seven crore fifty two lakh twenty one thousand three hundred two.
(d) Fifty eight million four hundred twenty three thousand two hundred two.
(e) Twenty three lakh thirty thousand ten.
3. Insert commas suitably and write the names according to Indian System of
Numeration :
(a) 87595762 (b) 8546283 (c) 99900046 (d) 98432701
4. Insert commas suitably and write the names according to International System
of Numeration :
(a) 78921092 (b) 7452283 (c) 99985102 (d) 48049831

13. KNOWING OUR NUMBERS
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(c) 1 kilometre = 1000 metres
How many millimetres make 1 kilometre?
Since 1 m = 1000 mm
1 km = 1000 m = 1000 × 1000 mm = 10,00,000 mm
We go to the market to buy rice or wheat; we buy it in
kilograms (kg). But items like ginger or chillies which
we do not need in large quantities, we buy in grams (g).
Weknow1kilogram=1000grams.
Have you noticed the weight of the medicine tablets
given to the sick? It is very small. It is in milligrams
(mg).
1 gram = 1000 milligrams.
What is the capacity of a bucket for holding water? It
is usually 20 litres ( ). Capacity is given in litres. But
sometimes we need a smaller unit, the millilitres.
A bottle of hair oil, a cleaning liquid or a soft drink
have labels which give the quantity of liquid inside in
millilitres (ml).
1 litre = 1000 millilitres.
Note that in all these units we have some words
common like kilo, milli and centi.You should remember
that among these kilo is the greatest and milli is the
smallest; kilo shows 1000 times greater, milli shows
1000 times smaller, i.e. 1 kilogram = 1000 grams,
1 gram = 1000 milligrams.
Similarly, centi shows 100 times smaller, i.e. 1 metre = 100 centimetres.
1. A bus started its journey and reached different places with a speed of
60 km/hour. The journey is shown below.
(i) Find the total distance covered by the bus from A to D.
(ii) Find the total distance covered by the bus from D to G.
(iii) Find the total distance covered by the bus, if it starts
from A and returns back to A.
(iv) Can you find the difference of distances from C to D and D to E?
1. How many
milligrams
make one
kilogram?
2. A box contains
2,00,000
medicine tablets
each weighing
20 mg. What is
the total weight
of all the
tablets in the
box in grams
and in
kilograms?

14. MATHEMATICS
14
The sales during the last year
Apples 2457 kg
Oranges 3004 kg
Combs 22760
Tooth brushes 25367
Pencils 38530
Note books 40002
Soap cakes 20005
(v) Find out the time taken by the bus to
reach
(a) A to B (b) C to D
(c) E to G (d) Totaljourney
2. Raman’s shop
Things Price
Apples Rs 40 per kg
Oranges Rs 30 per kg
Combs Rs 3 for one
Tooth brushes Rs 10 for one
Pencils Re 1 for one
Note books Rs 6 for one
Soap cakes Rs 8 for one
(a) CanyoufindthetotalweightofapplesandorangesRamansoldlastyear?
Weight of apples = __________ kg
Weight of oranges = _________ kg
Therefore, total weight = _____ kg + _____ kg = _____ kg
Answer – The total weight of oranges and apples = _________ kg.
(b) CanyoufindthetotalmoneyRamangotbysellingapples?
(c) Can you find the total money Raman got by selling apples and oranges
together?
(d) Make a table showing how much money Raman received from selling
each item. Arrange the entries of amount of money received in
descending order. Find the item which brought him the highest amount.
How much is this amount?

15. KNOWING OUR NUMBERS
15
We have done a lot of problems that have addition, subtraction, multiplication
and division. We will try solving some more here. Before starting, look at these
examples and follow the methods used.
Example 1 : Population of Sundarnagar was 2,35,471 in the year 1991. In the
year 2001 it was found to be increased by 72,958. What was the population of
the city in 2001?
Solution : Population of the city in 2001
= Population of the city in 1991 + Increase in population
= 2,35,471 + 72,958
Now, 235471
+ 72958
308429
Salma added them by writing 235471 as 200000 + 35000 + 471 and
72958as72000+958.Shegottheadditionas200000+107000+1429=308429.
Mary added it as 200000 + 35000 + 400 + 71 + 72000 + 900 + 58 = 308429
Answer : Population of the city in 2001 was 3,08,429.
All three methods are correct.
Example 2 : In one state, the number of bicycles sold in the year 2002-2003
was 7,43,000. In the year 2003-2004, the number of bicycles sold was 8,00,100.
In which year were more bicycles sold? and how many more?
Solution : Clearly, 8,00,100 is more than 7,43,000. So, in that state, more
bicycles were sold in the year 2003-2004 than in 2002-2003.
Now, 800100
– 743000
057100
Can you think of alternative ways of solving this problem?
Answer : 57,100 more bicycles were sold in the year 2003-2004.
Example 3 : The town newspaper is published every day. One copy has
12 pages. Everyday 11,980 copies are printed. How many total pages are
printed everyday?
Check the answer by adding
743000
+ 57100
800100 (the answer is right)

16. MATHEMATICS
16
Solution : Each copy has 12 pages. Hence, 11,980 copies will have
12 × 11,980 pages. What would this number be? More than 1,00,000 or lesser.
Try to estimate.
Now, 11980
× 12
23960
+ 119800
143760
Answer:Everyday 1,43,760 pages are printed.
Example 4 : The number of sheets of paper available for making notebooks
is 75,000. Each sheet makes 8 pages of a notebook. Each notebook contains
200 pages. How many notebooks can be made from the paper available?
Solution : Each sheet makes 8 pages.
Hence, 75,000 sheets make 8 × 75,000 pages,
Now, 75000
× 8
600000
Thus, 6,00,000 pages are available for making notebooks.
Now, 200 pages make 1 notebook.
Hence, 6,00,000 pages make 6,00,000 ÷ 200 notebooks.
3000
Now, 200 600000
– 600
0000 The answer is 3,000 notebooks.
EXERCISE 1.2
1. Abook exhibition was held for four days in a school. The number of tickets sold
at the counter on the first, second, third and final day was respectively 1094,
1812, 2050 and 2751. Find the total number of tickets sold on all the four days.
2. Shekhar is a famous cricket player. He has so far scored 6980 runs in test matches.
He wishes to complete 10,000 runs. How many more runs does he need?
3. In an election, the successful candidate registered 5,77,500 votes and his nearest
rival secured 3,48,700 votes. By what margin did the successful candidate win
the election?
4. Kirti bookstore sold books worth Rs 2,85,891 in the first week of June and books
worth Rs 4,00,768 in the second week of the month. How much was the sale for
)

17. KNOWING OUR NUMBERS
17
the two weeks together? In which week was the sale greater and by how much?
5. Findthedifferencebetweenthegreatestandtheleastnumberthatcanbewrittenusing
the digits 6, 2, 7, 4, 3 each only once.
6. Amachine, on an average, manufactures 2,825 screws a day. How many screws did
it produce in the month of January 2006?
7. Amerchant had Rs 78,592 with her. She placed an order for purchasing 40 radio sets
at Rs 1200 each. How much money will remain with her after the purchase?
8. A student multiplied 7236 by 65 instead of multiplying by 56. By how much was his
answer greater than the correct answer? (Hint: Do you need to do both the
multiplications?)
9. Tostitchashirt,2m15cmclothisneeded.Outof40mcloth,howmanyshirtscanbe
stitchedandhowmuchclothwillremain?
(Hint:convertdataincm.)
10. Medicineispackedinboxes,eachweighing4kg500g.Howmanysuchboxescanbe
loaded in a van which cannot carry beyond 800 kg?
11. The distance between the school and the house of a student’s house is 1 km 875 m.
Everyday she walks both ways. Find the total distance covered by her in six days.
12. A vessel has 4 litres and 500 ml of curd. In how many glasses, each of 25 ml
capacity, can it be filled?
1.3.1 Estimation
News
1. India drew with Pakistan in a hockey match watched by 51,000 spectators
in the stadium and 40 million television viewers world wide.
2. Approximately, 2000 people were killed and more than 50000 injured in a
cyclonic storm in coastal areas of India and Bangladesh.
3. Over 13 million passengers are carried over 63,000 kilometre route of
railway track every day.
Can we say that there were exactly as many people as the numbers quoted
in these news items? For example,
In (1), were there exactly 51,000 spectators in the stadium? or did exactly
40 million viewers watched the match on television?
Obviously, not. The word approximately itself
shows that the number of people were near about these
numbers. Clearly, 51,000 could be 50,800 or 51,300
but not 70,000. Similarly, 40 million implies much
more than 39 million but quite less than 41 million
but certainly not 50 million.

18. MATHEMATICS
18
The quantities given in the examples above are not exact counts, but are
estimates to give an idea of the quantity.
Discuss what each of these can suggest.
Where do we approximate? Imagine a big celebration at your home. The first
thingyoudoistofindoutroughlyhowmanyguestsmayvisityou.Canyougetan
ideaoftheexactnumberofvisitors?Itispracticallyimpossible.
Thefinanceministerofthecountrypresentsabudgetannually.Theministerprovides
forcertainamountunderthehead‘Education’.Cantheamountbeabsolutelyaccurate?
Itcanonlybeareasonablygoodestimateoftheexpenditurethecountryneedsfor
educationduringtheyear.
Thinkaboutthesituationswhereweneedtohavetheexactnumbersandcompare
themwithsituationswhereyoucandowithonlyanapproximatelyestimatednumber.
Givethreeexamplesofeachofsuchsituations.
1.3.2 Estimating to the nearest tens by rounding off
Lookatthefollowing:
(a) Findwhichflagsarecloserto260.
(b) Findtheflagswhicharecloserto270.
Locate the numbers 10,17 and 20 on your ruler. Is 17 nearer to 10 or 20? The
gap between 17 and 20 is smaller when compared to the gap between 17 and 10.
So, we round off 17 as 20, correct to the nearest tens.
Now consider 12, which also lies between 10 and 20. However, 12 is
closer to 10 than to 20. So, we round off 12 to 10, correct to the nearest tens.
How would you round off 76 to the nearest tens? Is it not 80?
We see that the numbers 1,2,3 and 4 are nearer to 0 than to 10. So, we
round off 1, 2, 3 and 4 as 0. Number 6, 7, 8, 9 are nearer to 10, so, we round
them off as 10. Number 5 is equidistant from both 0 and 10; it is a common
practice to round it off as 10.

19. KNOWING OUR NUMBERS
19
1.3.3 Estimating to the nearest hundreds by rounding off
Is 410 nearer to 400 or to 500?
410 is closer to 400, so it is rounded off to 400, correct to the nearest
hundred.
889 lies between 800 and 900.
It is nearer to 900, so it is rounded off as 900 correct to nearest hundred.
Numbers 1 to 49 are closer to 0 than to 100, and so are rounded off to 0.
Numbers 51 to 99 are closer to 100 than to 0, and so are rounded off to 100.
Number50isequidistantfrom0and100both.Itisacommonpracticetorounditoff
as100.
Checkifthefollowingroundingoffiscorrectornot:
841 800; 9537 9500; 49730 49700;
2546 2500; 286 200; 5750 5800;
168 200; 149 100; 9870 9800.
Correctthosewhicharewrong.
1.3.4 Estimating to the nearest thousands by rounding off
We know that numbers 1 to 499 are nearer to 0 than to 1000, so these numbers are
roundedoffas0.
The numbers 501 to 999 are nearer to 1000 than 0 so they are rounded off as
1000.
Number500isalsoroundedoffas1000.
Checkifthefollowingroundingoffiscorrectornot:
2573 3000; 53552 53000;
6404 6000; 65437 65000;
7805 7000; 3499 4000.
Correctthosewhicharewrong.
Round these numbers to the nearest tens.
28 32 52 41 39 48
64 59 99 215 1453 2936

20. MATHEMATICS
20
Round off the given numbers to the nearest tens, hundreds and thousands.
Given Number Approximate to Nearest Rounded Form
75847 Tens ________________
75847 Hundreds ________________
75847 Thousands ________________
75847 Ten thousands ________________
1.3.5 Estimating outcomes of number situations
How do we add numbers? We add numbers by following the algorithm (i.e. the
givenmethod)systematically.Wewritethenumberstakingcarethatthedigitsinthe
same place (ones, tens, hundreds etc.) are in the same column. For example,
3946 + 6579 + 2050 is written as —
Th H T O
3 9 4 6
6 5 7 9
+ 2 0 5 0
We add the column of ones and if necessary carry forward the appropriate
number to the tens place as would be in this case. We then add the tens
column and this goes on. Complete the rest of the sum yourself. This
procedure takes time.
There are many situations where we need to find answers more quickly.
For example, when you go to a fair or the market, you find a variety of attractive
things which you want to buy. You need to quickly decide what you can buy.
So, you need to estimate the amount you need. It is the sum of the prices of
things you want to buy.
Atrader is to receive money from two sources. The money he is to receive
is Rs 13,569 from one source and Rs 26,785 from another. He has to pay
Rs 37,000 to someone else by the evening. He rounds off the numbers to their
nearest thousands and quickly works out the rough answer. He is happy that
he has enough money.
Do you think he would have enough money? Can you tell without doing
the exact addition/subtraction?
Sheila and Mohan have to plan their monthly expenditure. They know
their monthly expenses on transport, on school requirements, on groceries,

21. KNOWING OUR NUMBERS
21
on milk, and on clothes and also on other regular
expenses. This month they have to go for visiting
and buying gifts. They estimate the amount they
would spend on all this and then add to see, if what
they have, would be enough.
Would they round off to thousands as the
trader did?
Think and discuss five more situations where we have to estimate sums or
remainders.
Did we use rounding off to the same place in all these?
There are no rigid rules when you want to estimate the outcomes of numbers.
The procedure depends on the degree of accuracy required and how quickly
the estimate is needed. The most important thing is, how sensible the guessed
answer would be.
1.3.6 To estimate sum or difference
As we have seen above we can round off a number to any place. The trader
rounded off the amounts to the nearest thousands and was satisfied that he had
enough. So, when you estimate any sum or difference, you should have an idea
of why you need to round off and therefore the place to which you would round
off. Look at the following examples.
Example 5 : Estimate: 5,290 + 17,986.
Solution : You find 17,986 > 5,290.
Round off to thousands.
17,986 is rounds off to 18,000
+5,290 is rounds off to + 5,000
Estimated sum = 23,000
Does the method work? You may attempt to find the actual answer and
verify if the estimate is reasonable.
Example 6 : Estimate: 5,673 – 436.
Solution : To begin with we round off to thousands. (Why?)
5,673 rounds off to 6,000
– 436 rounds off to – 0
Estimated difference = 6,000
This is not a reasonable estimate. Why is this not reasonable?

22. MATHEMATICS
22
To get a closer estimate, let us try rounding each number to hundreds.
5,673 rounds off to 5,700
– 436 rounds off to – 400
Estimated difference = 5,300
This is a better and more meaningful estimate.
1.3.7 To estimate products
How do we estimate a product?
What is the estimate for 19 × 78?
It is obvious that the product is less than 2000. Why?
If we approximate 19 to the nearest tens, we get 20 and then approximate 78
to nearest tens, we get 80 and 20 × 80 = 1600
Look at 63 × 182
If we approximate both to the nearest hundreds we
get 100 × 200 = 20,000. This is much larger than the
actual product. So, what do we do? To get a more
reasonable estimate, we try rounding off 63 to the
nearest 10, i.e. 60, and also 182 to the nearest ten, i.e.
180. We get 60 × 180 or 10,800. This is a good
estimate, but is not quick enough.
If we now try approximating 63 to 60 and 182 to
the nearest hundred, i.e. 200, we get 60 × 200, and this
number 12,000 is a quick as well as good estimate of
the product.
The general rule that we can make is, therefore, Round off each factor to its
greatest place, then multiply the rounded off factors. Thus, in the above
example, we rounded off 63 to tens and 182 to hundreds.
Now, estimate 81 × 479 using this rule :
479 is rounded off to 500 (rounding off to hundreds),
and 81 is rounded off to 80 (rounding off to tens).
The estimated product = 500 × 80 = 40,000
An important use of estimates for you will be to check your answers.
Suppose, you have done the multiplication 37 × 1889, but
arenotsureaboutyouranswer.Aquickandreasonableestimate
of the product will be 40 × 2000 i.e. 80,000. If your answer
is close to 80,000, it is probably right. On the other hand, if
it is close to 8000 or 8,00,000, something is surely wrong in
your multiplication.
Same general rule may be followed by addition and
subtraction of two or more numbers.
Estimate the
following products :
(a) 87 × 313
(b) 9 × 795
(c) 898 × 785
(d) 958 × 387
Make five more
such problems and
solve them.

23. KNOWING OUR NUMBERS
23
EXERCISE 1.3
1. Estimateeachofthefollowingusinggeneralrule:
(a) 730 + 998 (b) 796 – 314 (c) 12,904 +2,888 (d) 28,292 – 21,496
Maketenmoresuchexamplesofaddition,subtractionandestimationoftheiroutcome.
2. Givearoughestimate(byroundingofftonearesthundreds)andalsoacloserestimate
(by rounding off to nearest tens) :
(a) 439 + 334 + 4,317 (b) 1,08,734 – 47,599 (c) 8325 – 491
(d) 4,89,348 – 48,365
Make four more such examples.
3. Estimatethefollowingproductsusinggeneralrule:
(a) 578 × 161 (b) 5281 × 3491 (c) 1291 × 592 (d) 9250 × 29
Make four more such examples.
1.4 Using Brackets
Suman bought 6 notebooks from the market and the cost was Rs 10 per notebook.
Her sister Sama also bought 7 notebooks of the same type. Find the total money
they paid.
Seema calculated the Meera calculated the
amount like this amount like this
6 × 10 + 7 × 10 6 + 7 =13
= 60 + 70
= 130 and 13 × 10 = 130
Ans. Rs 130 Ans. Rs 130
1. Writetheexpressionsforeachofthefollowingusingbrackets.
(a) Four multiplied by the sum of nine and two.
(b) Divide the difference of eighteen and six by four.
(c) Forty five divided by three times the sum of three and two.
2. Write three different situations for (5 + 8) × 6.
(One such situation is : Sohani and Reeta work for 6 days; Sohani
works 5 hours a day and Reeta 8 hours a day. How many hours do
both of them work in a week?)
3. Write five situations for the following where brackets would be
necessary. (a) 7(8 – 3) (b) (7 + 2) (10 – 3)

24. MATHEMATICS
24
YoucanseethatSeema’sandMeera’swaystogettheanswerareabitdifferent.
But both give the correct result. Why?
Seema says, what Meera has done is 7 + 6 × 10.
Appu points out that 7 + 6 × 10 = 7 + 60 = 67. Thus, this is not what Meera
had done.All the three students are confused.
Toavoidconfusioninsuchcaseswemayusebrackets.Wecanpackthenumbers
6and7togetherusingabracket,indicatingthatthepackistobetreatedasasingle
number. Thus, the answer is found by (6 + 7) × 10 = 13 × 10.
This is what Meera did. She first added 6 and 7 and then multiplied the sum
by 10.
This clearly tells us : First, turn everything inside the brackets ( ) into a
single number and then do the operation outside which in this case is to
multiply by 10.
1.4.1 Expanding brackets
Now, observe how use of brackets allows us to follow our procedure
systematically. Do you think that it will be easy to keep a track of what steps we
have to follow without using brackets?
(i) 7 × 109 = 7 × (100 + 9) = 7 × 100 + 7 × 9 = 700 + 63 = 763
(ii)102 × 103 = (100 + 2) × (100 + 3) = (100 + 2) × 100 + (100 + 2) × 3
= 100 × 100 + 2 × 100 + 100 × 3 + 2 × 3
= 10,000 + 200 + 300 + 6 = 10,000 + 500 + 6
= 10,506
(iii) 17 × 109 = (10 + 7) × 109 = 10 × 109 + 7 × 109
= 10 × (100 + 9) + 7 × (100 + 9)
= 10 × 100 + 10 × 9 + 7 × 100 + 7 × 9
= 1000 + 90 + 700 + 63 = 1,790 + 63
= 1,853
1.5 Roman Numerals
We have been using the Hindu-Arabic numeral system so far. This
is not the only system available. One of the early systems of
writing numerals is the system of Roman numerals. This system
is still used in many places.
For example, we can see the use of Roman numerals in clocks; it
is also used for classes in the school time table etc.
Find three other examples, where Roman numerals are used.
The Roman numerals
XII
I
II
III
IV
V
VI
VII
VIII
IX
X
XI

25. KNOWING OUR NUMBERS
25
I, II, III, IV, V, VI, VII, VIII, IX, X
denote 1,2,3,4,5,6,7,8,9 and 10 respectively. This is followed by XI for 11, XII
for 12,... till XX for 20. Some more Roman numerals are :
I V X L C D M
1 5 10 50 100 500 1000
The rules for the system are :
(a) If a symbol is repeated, its value is added as many times as it occurs:
i.e. II is equal 2, XX is 20 and XXX is 30.
(b) Asymbol is not repeated more than three times. But the symbols V, Land D
are never repeated.
(c) If a symbol of smaller value is written to the right of a symbol of greater
value, its value gets added to the value of greater symbol.
VI = 5 + 1 = 6, XII = 10 + 2 = 12
and LXV = 50 + 10 + 5 = 65
(d) If a symbol of smaller value is written to the left of a symbol of greater
value, its value is subtracted from the value of the greater symbol.
IV = 5 – 1 = 4, IX = 10 – 1 = 9
XL= 50 – 10 = 40, XC = 100 – 10 = 90
(e) The symbols V, L and D are never written to the left of a symbol of greater
value, i.e.V, Land D are never subtracted.
The symbol I can be subtracted from V and X only.
The symbol X can be subtracted from L, M and C only.
Following these rules we get,
1 = I 10 = X 100 = C
2 = II 20 = XX
3 = III 30 = XXX
4 = IV 40 = XL
5 = V 50 = L
6 = VI 60 = LX
7 = VII 70 = LXX
8 = VIII 80 = LXXX
9 = IX 90 = XC
(a) Write in Roman numerals the missing numbers in the
table.
(b) XXXX, VX, IC, XVV are not written. Can you tell why?
Write in
Roman
numerals.
1. 73
2. 92

26. MATHEMATICS
26
Example 7 : Write in Roman Numerals (a) 69 (b) 98.
Solution : (a) 69 = 60 + 9 (b) 98 = 90 + 8
= (50 + 10) + 9 = (100 – 10) + 8
= LX + IX = XC + VIII
= LX IX =XCVIII
What have we discussed?
1. Given two numbers, one with more digits is the greater number. If the number of
digits in two given numbers is the same, that number is larger, which has a greater
leftmost digit. If this digit also happens to be the same, we look at the next digit and
so on.
2. In forming numbers from given digits, we should be careful to see if the conditions
under which the numbers are to be formed are satisfied. Thus, to form the greatest
four digit number from 7, 8, 3, 5 without repeating a single digit, we need to use all
fourdigits,thegreatestnumbercanhaveonly8astheleftmostdigit.
3. The smallest four digit number is 1000 (one thousand). It follows the largest three
digitnumber999.Similarly,thesmallestfivedigitnumberis10,000.Itistenthousand
andfollowsthelargestfourdigitnumber9999.
Further,thesmallestsixdigitnumberis100,000.Itisonelakhandfollowsthelargest
fivedigitnumber99,999.Thiscarriesonforhigherdigitnumbersinasimilarmanner.
4. Use of commas helps in reading and writing large numbers. In the Indian system of
numerationwehavecommasafter3digitsstartingfromtherightandthereafterevery
2 digits. The commas after 3, 5 and 7 digits separate thousand, lakh and crore
respectively.IntheInternationalsystemofnumerationcommasareplacedafterevery
3 digits starting from the right. The commas after 3 and 6 digits separate thousand
andmillionrespectively.
5. Largenumbersareneededinmanyplacesindailylife.Forexample,forgivingnumber
ofstudentsinaschool,numberofpeopleinavillageortown,moneypaidorreceived
in large transactions (paying and selling), in measuring large distances say betwen
various cities in a country or in the world and so on.
6. Remember kilo shows 1000 times larger, Centi shows 100 times smaller and milli
shows1000timessmaller,thus,1kilometre=1000metres,1metre=100centimetres
or1000millimetresetc.
7. Thereareanumberofsituationsinwhichwedonotneedtheexactquantitybutneed
onlyareasonableguessoranestimate.Forexample,whilestatinghowmanyspectators
watchedaparticularinternationalhockeymatch,westatetheapproximatenumber,
say 51,000, we do not need to state the exact number.

27. KNOWING OUR NUMBERS
27
8. Estimation involves approximating a quantity to an accuracy required.Thus, 4117
may be approximated to 4100 or to 4000, i.e. to the nearest hundred or to the
nearest thousand depending on our need.
9. Innumberofsituations,wehavetoestimatetheoutcomeofnumberoperations.This
isdonebyroundingoffthenumbersinvolvedandgettingaquick,roughanswer.
10. Estimatingtheoutcomeofnumberoperationsisusefulincheckinganswers.
11. Use of brackets allows us to avoid confusion in the problems where we need to
carry out more than one number operation.
12. WeusetheHindu-Arabicsystemofnumerals.Anothersystemofwritingnumeralsis
theRomansystem.